MR2071328
Carrasco, Pilar(E-GRAN-AL); Martínez-Moreno, Juan(E-JAE)
Simplicial cohomology with coefficients in symmetric categorical groups. (English. English summary)
Appl. Categ. Structures 12 (2004), no. 3, 257--285.
18D10 (18G50 18G60)

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{A review for this item is in process.}

MR2068776
Martínez-Moreno, J.(E-JAE); Quesada, J. M.(E-JAE)
Shape preserving properties and eigenstructure of Bleimann, Butzer and Hahn operators. (English. English summary)
Int. J. Differ. Equ. Appl. 8 (2003), no. 2, 115--127.
41A36

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{A review for this item is in process.}


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MR2026041 (2004j:41038)
López-Moreno, A. J.(E-JAE); Martínez-Moreno, J.(E-JAE); Muñoz-Delgado, F. J.(E-JAE)
Asymptotic behavior of Kantorovich type operators. (English. English summary)
Seventh Zaragoza-Pau Conference on Applied Mathematics and Statistics (Spanish) (Jaca, 2001), 399--404,
Monogr. Semin. Mat. García Galdeano, 27,
Univ. Zaragoza, Zaragoza, 2003.
41A36 (41A35)

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The paper deals with a generalization of the Kantorovich operators introduced by D. P. Kacsó \ref[Rend. Circ. Mat. Palermo (2) Suppl. 2002, no. 68, part II, 523--538; MR1975467 (2004c:41029)]. If $S^{q}$ defines the $q$-th iteration of the integral operator $Sf\left(x\right)=\int_{0}^{x}f\left(z\right)dz$, then the linear $q$-th Kantorovich operators are defined by $K_{q,n}=D^{q}B_{n+1}S^{q}$, where $D^{q}$ denotes the derivative of order $q$, and $B_{n+1}$ is the linear positive operator given by the Bernstein polynomial of degree $n+1$.

The authors establish asymptotic expressions, expansions of Voronovskaja type and a saturation theorem for all derivatives of the $q$-th Kantorovich operators $K_{q,n}$.

{For the entire collection see MR2025196 (2004h:00021).}

Reviewed by Petru P. Blaga
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MR2013063 (2004i:41029)
Quesada, J. M.(E-JAEHP); Martínez-Moreno, J.(E-JAEHP); Navas, J.(E-JAEHP); Fernández-Ochoa, J.
Taylor expansion of best $l\sb p$-approximations about $p=1$. (English. English summary)
Appl. Math. Lett. 16 (2003), no. 7, 993--998.
41A58 (41A50)

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In this paper, the authors consider the problem of best approximation in $l_p(n), 1\leq p\leq\infty$; that is, the space $\Bbb R^n$ with the $l_p$-norms. For any non-empty set $K\subset\Bbb R^n$ and $h\in \Bbb R^n\sbs K$, an element $h_p\in K$ is called a best $l_p$-approximation of $h$ from $K$ if $\|h_p-h\|_p\leq\|f-h\|_p$ for all $f\in K$. It is known that if $K$ is an affine subspace of $\Bbb R^n$, then $\lim_{p\rightarrow 1}h_p=h_1^{*}$, where $h_1^{*}$ is the best $l _1$-approximation of $0$ from $K$. The above convergence is usually called the Polya $1$-algorithm. A. G. Egger and G. D. Taylor \ref[J. Approx. Theory 75 (1993), no. 3, 312--324; MR1250543 (94m:41044)] have proved that $\|h_p-h_1^{*}\|/(p-1)$ is bounded as $p\rightarrow 1^{+}$; further, if $h_1^{*}$ is unique, $\|h_p-h_1^{*}\|=\scr O(\gamma^{1/(p-1)})$ for some $0\leq \gamma<1$. In a recent paper [J. Approx. Theory 118 (2002), no. 2, 316--329; MR1932583 (2003i:41036)], the authors gave a complete description of the rate of convergence of $h_p$ to $h_1^{*}$ as $p\rightarrow 1$. Here, the authors prove that the best $l_p$-approximation, $h_p$, has a Taylor expansion of order $r$ about $p=1$ of the type $6h_p=h_1^{*}+\sum_{i=1}^r\alpha_l(p-1)^l+\gamma_p^{(r)}$, for some $\alpha_l\in\Bbb R^n, 1\leq l\leq r$, and $\gamma_p^{(r)}\in \Bbb R^n$ with $\|\gamma_p^{(r)}\|=\scr{O}((p-1)^{r+1}))$.

Reviewed by Liaqat Ali Khan
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MR1975473 (Review)
López-Moreno, Antonio-Jesús(E-JAE); Martínez-Moreno, Juan(E-JAE); Muñoz-Delgado, Francisco-Javier(E-JAE)
Asymptotic expression of derivatives of Bernstein type operators. (English. English summary)
Proceedings of the Fourth International Conference on Functional Analysis and Approximation Theory, Vol. II (Potenza, 2000).
Rend. Circ. Mat. Palermo (2) Suppl. 2002, no. 68, part II, 615--624.
41A35

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Summary: "We present several techniques to get complete asymptotic expressions for the derivatives of Bernstein type operators even if they do not preserve classical shape properties such as the $i$-convexities."
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MR1966067
López-Moreno, A. J.(E-JAE); Martínez-Moreno, J.(E-JAE); Muñoz-Delgado, F. J.(E-JAE); Quesada, J. M.(E-JAE)
Some properties of linear positive operators defined in terms of finite differences. (English. English summary)
Multivariate approximation and interpolation with applications (Almuñécar, 2001), 87--96,
Monogr. Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, 20,
Acad. Cienc. Exact. Fís. Quím. Nat. Zaragoza, Zaragoza, 2002.
41A36 (41A30)

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{This item will not be reviewed individually. For details of the collection in which this item appears see MR1966061 (2003j:41001) .}


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MR1932583 (2003i:41036)
Quesada, J. M.(E-JAE); Martínez-Moreno, J.(E-JAE); Navas, J.(E-JAE); Fernández-Ochoa, J.
Rate of convergence of the linear discrete Pólya 1-algorithm. (English. English summary)
J. Approx. Theory 118 (2002), no. 2, 316--329.
41A50 (41A25)

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The problem of best approximation in $l_p(n), 1\le p\le\infty$, is considered. Let $K\ne 0$ be a subset of $\bold R^n$. For $p>1$ the element $h_p\in K$ is a best $l_p$-approximation of $h$ if $\|h_p-h\|_p\le\|f-h\|_p$ for all $f\in K$. It is known that if $K$ is an affine subspace of $\bold R^n$, then $\lim_{p\to 1} h_p = h_1^*,$ where $h_1^*$ is a best $l_1$-approximation of $h$ from $K$ called the natural best $l_1$-approximation. A complete description of the rate of convergence of $h_p$ to $h_1^*,$ as $p\to 1$, is given.

Reviewed by Yuri V. Kryakin
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[References]

1. A. G. Egger and G. D. Taylor, Rate of convergence of the discrete Pólya-1 algorithm, J. Approx. Theory 75 (1993), 312--324. MR1250543 (94m:41044)
2. J. Fischer, The convergence of the best discrete linear $L_p$ approximation as $p \rightarrow 1$, J. Approx. Theory 39 (1983), 374--385. MR0723228 (85d:41027)
3. D. Landers and L. Rogge, Natural choice of $L_1$-approximants, J. Approx. Theory 33 (1981), 268--280. MR0647853 (83f:41021)
4. A. Pinkus, "On $L_1$-Approximation," Cambridge Univ. Press, Cambridge, UK, 1989.
5. J. M. Quesada and J. Navas, Rate of convergence of the linear discrete Polya algorithm, J. Approx. Theory 110 (2001), 109--119. MR1826088 (2002b:41025)
6. I. Singer, "Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces," Springer-Verlag, Berlin, 1970. MR0270044 (42 #4937)

MR1932580 (2003g:41047)
Quesada, J. M.(E-JAE); Martínez-Moreno, J.(E-JAE); Navas, J.(E-JAE)
Asymptotic behaviour of best $l\sb p$-approximations from affine subspaces. (English. English summary)
J. Approx. Theory 118 (2002), no. 2, 275--289.
41A50 (41A25)

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The problem of best approximation of an element $h\in\Bbb R^n$ by elements of an affine subspace $K$ of $\Bbb R^n$ with respect to the $l_p$-norm, $1<p\le\infty$, is considered. It is well known \ref[see, e.g., J. Descloux, J. Soc. Indust. Appl. Math. 11 (1963), 1017--1026; MR0159172 (28 \#2389)] that if $h_p, 1<p<\infty$, denotes the unique best $l_p$-approximation of $h$ from $K$, then $$\lim_{p\to\infty}h_p=h_\infty^*$$ where $h_\infty^*$ is a best uniform approximation, the so-called strict uniform approximation of $h$.

In this paper it is shown that for all $r\in\Bbb N$ and $1<p<\infty$ there exist $\gamma_p^{(r)}, \alpha_j\in\Bbb R^n, 1\le j\le r$, such that $$h_p=h_\infty^*+\frac{\alpha_1}{p-1}+\frac{\alpha_2}{(p-1)^2}+\dots+ \frac{\alpha_r}{(p-1)^r}+\gamma_p^{(r)}$$ with $\|\gamma_p^{(r)}\|=O(p^{-r-1})$.

Reviewed by Manfred Sommer
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[References]

1. J. Descloux, Approximations in $L^p$ and Chebychev approximations, J. Soc. Ind. Appl. Math. 11 (1963), 1017--1026. MR0159172 (28 #2389)
2. A. Egger and R. Huotari, Rate of convergence of the discrete Polya algorithm, J. Approx. Theory 60 (1990), 24--30. MR1028891 (91c:41070)
3. R. Fletcher, J. A. Grant, and M. D. Hebden, Linear minimax approximation as the limit of best $L_p$-approximation, SIAM J. Numer. Anal. 11 (1974), 123--136. MR0343535 (49 #8276)
4. M. Marano, Strict approximation on closed convex sets, Approx. Theory Appl. 6 (1990), 99--109. MR1070617 (91g:41040)
5. M. Marano and J. Navas, The linear discrete Polya algorithm, Appl. Math. Lett. 8 (1995), 25--28. MR1368034
6. J. M. Quesada and J. Navas, Rate of convergence of the linear discrete Polya algorithm, J. Approx. Theory 110 (2001), 109--119. MR1826088 (2002b:41025)
7. J. R. Rice, Thebychefff approximation in a compact metric space, Bull. Amer. Math. Soc. 68 (1962), 405--410. MR0139886 (25 #3313)
8. I. Singer, "Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces," Springer-Verlag, Berlin, 1970. MR0270044 (42 #4937)

MR1932886 (2003g:41046)
Quesada, J. M.(E-JAEUP); Martínez-Moreno, J.(E-JAEUP); Navas, J.(E-JAEUP); Fernández-Ochoa, J.
Rate of convergence of the best $l\sb p$-approximations from affine subspaces as $p\to1$. (English. English summary)
C. R. Acad. Bulgare Sci. 55 (2002), no. 9, 13--18.
41A50 (41A25)

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Summary: "In this paper we consider the problem of best approximation in $l_p(n)$, $1\leq p\leq\infty$. If $h_p, 1\leq p<\infty$, denotes the best $l_p$-approximation of the element $h\in\Bbb R^n$ from a proper affine subspace $K$ of $\Bbb R^n$, $h\notin K$, then $\lim_{p\to 1}h_p=h^\ast_1$, where $h^\ast_1$ is a best $l_1$-approximation of $h$ from $K$, the so-called natural $l_1$-approximation. Our aim is to give a complete description of the rate of convergence of $h_p$ to $h^\ast_1$ as $p\to 1$."
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MR1924865 (2003e:41038)
López-Moreno, Antonio-Jesús(E-JAE); Martínez-Moreno, J.(E-JAE); Muñoz-Delgado, F.-J.(E-JAE)
Asymptotic expressions for multivariate positive linear operators. (English. English summary)
Approximation theory, X (St. Louis, MO, 2001), 287--307,
Innov. Appl. Math.,
Vanderbilt Univ. Press, Nashville, TN, 2002.
41A36 (41A60)

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The authors prove general results for multivariate positive linear operators in order to compute the asymptotic expansion of the partial derivatives of the operators discussed. In this way, the two-dimensional version of a classical result due to Ulrich Abel is extended to an arbitrary dimension. Starting from the work of Abel (seven papers of his are being quoted), the present paper deals in a unitary, elegant and complete manner with an asymptotic expression of convex operators of order $\alpha $ $(\alpha \in\Bbb{N}_0^m)$ in the multivariate case, which is different from the tensor product construction. The main tools used are Leibniz's multivariate formula and Taylor's formula. These new results make it possible to study the asymptotic behaviour of the partial derivatives of the analyzed sequence of linear positive operators satisfying certain natural conditions.

As for applications, the authors study both the multivariate Baskakov-Schurer operator and the Bernstein operators on a simplex. Let $|·|$ denote the sum of the components for elements of $\Bbb{R}^m$ and ${a\choose k}$ stand for $\frac{a!}{k!(a-|k|)!}$ $(a\in\Bbb{N}_0, k\in\Bbb{N}_0^m)$. The first operator is defined for all functions $f\in {\scr D}$ and $x\in H\coloneq (\Bbb{R}_0^+)^m$ as follows: $$\multline A_{n,p}f(x)=\\ (1+|x|)^{-(n+p)}\sum_{k=0}^\infty {n+p+|k|-1\choose k}\left(\frac{x}{1+|x|}\right)^k f\left(\frac{k}{n}\right),\endmultline$$ where $n\in\Bbb{N}$, $p\in \Bbb{N}_0$, and ${\scr D}=\{f\in \Bbb{R}^H\colon \forall x\in H,\ \forall n\in \Bbb{N}, |A_{n,p}f(x)|<\infty \}$. The second operator studied has the form $$B_n f(x)=\sum_{k\in \Bbb{N}_0^m,|k|\le n}{n\choose k} x^k(1-|x|)^{n-|k|}f\left(\frac{k}{n}\right),$$ where $n\in\Bbb{N}$, $x\in S\coloneq \{x\in \Bbb{R}^m\colon \ |x|\le 1, 0\le x\}$ and $f\in \Bbb{R}^S$.

At the same time, the authors establish Voronovskaya-type formulas for all partial derivatives of the above-mentioned operators.

{For the entire collection see MR1924847 (2003c:41001).}

Reviewed by Octavian Agratini
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MR1911650 (2003b:41025)
Quesada, José María(E-JAE); Martínez-Moreno, Juan(E-JAE); Navas, Juan(E-JAE)
On best $p$-approximation from affine subspaces: asymptotic expansion. (English. English, French summary)
C. R. Math. Acad. Sci. Paris 334 (2002), no. 12, 1077--1082.
41A60 (41A50 46N10)

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Summary: "In this paper we consider the problem of best approximation in $l_p(n)$, $1<p\leq\infty$. If $h_p, 1<p<\infty$, denotes the best $p$-approximation of the element $h\in\Bbb R^n$ from a proper affine subspace $K$ of $\Bbb R^n$, $h\notin K$, then $\lim_{p\to\infty}h_p=h^\ast_\infty$, where $h^\ast_\infty$ is a best uniform approximation of $h$ from $K$, the so-called strict uniform approximation. Our aim is to prove that for all $r\in\Bbb N$ there are $\alpha_j\in\Bbb R^n$, $1\leq j\leq r$, such that $$h_p=h^\ast_\infty+\frac{\alpha_1}{p-1}+\frac{\alpha_2}{(p-1)^2}+\dots+\frac{\alpha_r}{(p-1)^r}+\gamma^{(r)}_p,$$ with $\gamma^{(r)}_p\in\Bbb R^n$ and $\|\gamma^{(r)}_p\|=O(p^{-r-1})$."
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MR1899770
López-Moreno, A. J.(E-JAE); Martínez-Moreno, J.(E-JAE); Muñoz-Delgado, F. J.(E-JAE)
Eigenvectors of conservative operators. (Spanish. Spanish summary)
Proceedings of the Meeting of Andalusian Mathematicians, Vol. II (Spanish) (Sevilla, 2000), 625--632,
Colecc. Abierta, 52,
Univ. Sevilla Secr. Publ., Seville, 2001.
41A35

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{This item will not be reviewed individually. For details of the collection in which this item appears see MR1899701 (2003a:00034) .}


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MR1875264
López-Moreno, Antonio-Jesús(E-JAE); Martínez-Moreno, Juan(E-JAE); Muñoz-Delgado, Francisco-Javier(E-JAE)
Asymptotic expressions for positive linear operators defined on multivalued functions. (Spanish. English summary)
Actes des VI$\sp {{\rm \grave emes}}$ Journées Zaragoza-Pau de Mathématiques Appliquées et de Statistiques (Jaca, 1999), 393--400,
Publ. Univ. Pau, Pau, 2001.
41A60

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{This item will not be reviewed individually. For details of the collection in which this item appears see MR1875217 (2002h:00018) .}


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MR1875263
López-Moreno, Antonio-Jesús(E-JAE); Martínez-Moreno, Juan(E-JAE); Muñoz-Delgado, Francisco-Javier(E-JAE)
Expressing some Bernstein operators via nonpolynomial test functions. (Spanish. English summary)
Actes des VI$\sp {{\rm \grave emes}}$ Journées Zaragoza-Pau de Mathématiques Appliquées et de Statistiques (Jaca, 1999), 385--392,
Publ. Univ. Pau, Pau, 2001.
41A35

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{This item will not be reviewed individually. For details of the collection in which this item appears see MR1875217 (2002h:00018) .}


Previous Item

MR1852118 (2003a:18007)
Carrasco, Pilar(E-GRAN-AL); Martínez Moreno, J.(E-JAE)
Categorical $G$-crossed modules and 2-fold extensions. (English. English summary)
J. Pure Appl. Algebra 163 (2001), no. 3, 235--257.
18D10 (18G50 18G60)

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This interesting paper gives an interpretation, in terms of certain $2$-fold extensions, of the fourth cohomology group $H^4(G,{\Bbb A})$ of a group $G$ with coefficients in a symmetric $G$-categorical group $\Bbb A$. Here $\Bbb A$ is a braided monoidal category such that every arrow is invertible; the functors $Y\mapsto X\otimes Y$, $Y\mapsto Y\otimes X$ are equivalences for each object $X$; the braiding $$c=c_{X,Y}\colon X\otimes Y \rightarrow Y\otimes X$$ satisfies $c^2=1$; there are equivalences ${^g(-)}\colon {\Bbb A} \rightarrow {\Bbb A}, X \mapsto {^gX}$ for each $g\in G$ together with natural isomorphisms $${^g(X\otimes Y)} \rightarrow ({^gX}\otimes {^gY}),\ {^{g+h}X} \rightarrow {^g(^hX)}, {^0X}\rightarrow X.$$ The cohomology group is that defined by K.-H. Ulbrich \ref[J. Algebra 91 (1984), no. 2, 464--498; MR0769585 (86h:18003)]. If $\Bbb A$ is strictly coherent then it is a Picard category with a $G$-module structure in the sense of Ulbrich. The structure ${\Bbb A}$ is also related to the notion of monoidal $G$-graded category due to A.\,Fröhlich and C. T. C. Wall. An ordinary ${\Bbb Z}G$-module $A$ can be viewed either as a $G$-categorical group $A_1$ with one object, or as a discrete $G$-categorical group $A_0$, and there are isomorphisms $H^4(G,A_1)\cong H^4_{E-M}(G,A)$, $H^4(G,A_0)\cong H^3_{E-M}(G,A)$ relating Ulbrich's cohomology with Eilenberg-Mac\,Lane cohomology.

The authors define the notion of a categorical $G$-crossed module and the related notion of a $2$-fold extension of $G$ by $\Bbb A$. When ${\Bbb A}=A_0$ the latter coincides with the usual notion of $2$-fold extension of a group $G$ by a ${\Bbb Z}G$-module $A$. When ${\Bbb A}=A_1$ one recovers the notion of a nonabelian $3$-extension of $G$ by $A$ in the sense of D. Conduché \ref[J. Pure Appl. Algebra 34 (1984), no. 2-3, 155--178; MR0772056 (86g:20068)]. The main result is a bijection between $H^4(G,{\Bbb A})$ and the set of equivalence classes of $2$-fold extensions of $G$ by $\Bbb A$. This generalizes the well-known interpretation of $H^3_{E-M}(G,A)$. It also generalizes Conduché's interpretation of $H^4(G,A)$. The proof involves the explicit construction of a $4$-cocycle from a $2$-fold extension.

Reviewed by Graham J. Ellis
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MR1627474 (99h:18014)
Carrasco, P.(E-GRANS-AL); Martínez Moreno, J.(E-JAEUP)
A $2$-dimensional cohomology with coefficients in categorical groups.
Extracta Math. 12 (1997), no. 3, 231--241.
18G55 (18D10 55P20 55U10)

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The homotopy category of topological spaces with non-vanishing homotopy groups only in dimensions 1 and 2 is known to be equivalent to a category of purely algebraic objects, known as categorical groups. The paper under review defines a set-valued cohomology functor with values in a categorical group, and proves that it is naturally equivalent to the set of homotopy classes of maps to the corresponding topological space. This is analogous to, and extends, the relationship between the standard cohomology functor and Eilenberg-Mac Lane spaces.

Reviewed by P. S. Green
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